scholarly journals Calculation of signal spectrum by means of stochastic inversion

2010 ◽  
Vol 28 (7) ◽  
pp. 1409-1418 ◽  
Author(s):  
T. Nygrén ◽  
Th. Ulich
Keyword(s):  
Fourier Transform ◽  
Sampling Rate ◽  
Digital Signal ◽  
Close Relation ◽  
Signal Spectrum ◽  
Inversion Method ◽  
Stochastic Inversion ◽  
Data Points ◽  
Sinusoidal Functions ◽  
The Fourier Transform

Abstract. The standard method of calculating the spectrum of a digital signal is based on the Fourier transform, which gives the amplitude and phase spectra at a set of equidistant frequencies from signal samples taken at equal intervals. In this paper a different method based on stochastic inversion is introduced. It does not imply a fixed sampling rate, and therefore it is useful in analysing geophysical signals which may be unequally sampled or may have missing data points. This could not be done by means of Fourier transform without preliminary interpolation. Another feature of the inversion method is that it allows unequal frequency steps in the spectrum, although this property is not needed in practice. The method has a close relation to methods based on least-squares fitting of sinusoidal functions to the signal. However, the number of frequency bins is not limited by the number of signal samples. In Fourier transform this can be achieved by means of additional zero-valued samples, but no such extra samples are used in this method. Finally, if the standard deviation of the samples is known, the method is also able to give error limits to the spectrum. This helps in recognising signal peaks in noisy spectra.

scholarly journals Some advantages of non-binary Galois fields for digital signal processing

2021 ◽  
Vol 23 (2) ◽  
pp. 871
Author(s):  
Inabat Moldakhan ◽  
Dinara K. Matrassulova ◽  
Dina B. Shaltykova ◽  
Ibragim E. Suleimenov
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Digital Signal ◽  
Galois Field ◽  
Prime Numbers ◽  
Sampling Interval ◽  
Signal Spectrum ◽  
Galois Fields ◽  
Digital Spectrum ◽  
The Fourier Transform

It is shown that the convenient processing facilities of digital signals that varying in a finite range of amplitudes are non-binary Galois fields, the numbers of which elements are equal to prime numbers. Within choosing a sampling interval which corresponding to such a Galois field, it becomes possible to construct a Galois field Fourier transform, a distinctive feature of which is the exact correspondence with the ranges of variation of the amplitudes of the original signal and its digital spectrum. This favorably distinguishes the Galois Field Fourier Transform of the proposed type from the spectra, which calculated using, for example, the Walsh basis. It is also shown, that Galois Field Fourier Transforms of the proposed type have the same properties as the Fourier transform associated with the expansion in terms of the basis of harmonic functions. In particular, an analogue of the classical correlation, which connected the signal spectrum and its derivative, was obtained. On this basis proved, that the using of the proposed type of Galois fields makes it possible to develop a complete analogue of the transfer function apparatus, but only for signals presented in digital form.

APPLICATION OF KOTELNIKOV’S THEOREM TO ANALYSIS OF RANDOM PROCESSES

2019 ◽  
pp. 28-34
Author(s):  
O. V. Goriunov ◽  
S. V. Slovtsov
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Random Process ◽  
Spectral Characteristics ◽  
Random Processes ◽  
Statistical Characteristics ◽  
Signal Spectrum ◽  
The Fourier Transform ◽  
The Stability ◽  
Correlation And Spectral Analysis

Analysis of many dynamic tasks arising in engineering applications is associated with the construction of spectral characteristics. However, the application of spectral analysis to random oscillations, which in most cases describe real processes (technical, technological, etc.), has a number of features and limitations associated, in particular, with the anconvergence of the Fourier transform. The substantiated metrological evaluation of the spectra associated with the reliability of the applied results is complicated by the absence of a rigorous mathematical model of a random process. The above remarks were solved on the basis of application of Kotelnikov's theorem at decomposition of a random process on known eigenfunctions. The obtained decomposition allowed us to obtain a number of results in the field of correlation and spectral analysis of random processes: the stability of the ACF and the relationship with the statistical characteristics of the implementation is proved, the orthogonal decomposition of the random process in the form of a continuous function is presented, which allows us to consider the evaluation and analyze the characteristics of the realizations without the use of a fast Fourier transform; the natural relationship between ACF and spectral density for a time-limited signal is shown, and the symmetric form of recording the signal spectrum is justified.

The λ-cosine transforms with odd kernel and the hemispherical transform

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Fourier Transform ◽  
Unit Sphere ◽  
Borel Measure ◽  
Fractional Integrals ◽  
Inversion Method ◽  
Spherical Means ◽  
Finite Borel Measure ◽  
Basic Facts ◽  
Hemispherical Transform ◽  
The Fourier Transform

AbstractWe review some basic facts about the λ-cosine transforms with odd kernel on the unit sphere S n−1 in ℝn. These transforms are represented by the spherical fractional integrals arising as a result of evaluation of the Fourier transform of homogeneous functions. The related topic is the hemispherical transform which assigns to every finite Borel measure on S n−1 its values for all hemispheres. We revisit the known facts about this transform and obtain new results. In particular, we show that the classical Funk- Radon-Helgason inversion method of spherical means is applicable to the hemispherical transform of L p-functions.

GRAVITY PROFILE INTERPRETATION USING THE FOURIER TRANSFORM

Geophysics ◽  
1974 ◽  
Vol 39 (6) ◽  
pp. 862-866 ◽  
Author(s):  
S. J. Collins ◽  
A. R. Dodds ◽  
B. D. Johnson
Keyword(s):  
Fourier Transform ◽  
Horizontal Cylinder ◽  
Direct Interpretation ◽  
Data Points ◽  
Lateral Extent ◽  
The Fourier Transform ◽  
Interpretation Method ◽  
Data Length ◽  
Gravity Profile

A number of attempts have been made to perform direct interpretation of gravity profiles using the Fourier transform of the profile. Of these, the methods of Odegard and Berg (1965) and Sharma et al. (1970) appear to be most applicable. The purpose of this study was to take one of the proposed models (Odegard and Berg’s horizontal cylinder) and determine the applicability of the interpretation method in terms of the number and lateral extent of the data points. The relative accuracies of the estimates of the depth and mass of a cylinder were determined as criteria for estimating the effects of data length and number of data points. In addition, the interpretation was extended to include the separation of two cylinders.

scholarly journals Algorithm for B-scan Image Reconstruction in Optical Coherence Tomography

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kranti Patil ◽  
Anurag Mahajan ◽  
Balamurugan Subramani ◽  
Arulmozhivarman Pachiyappan ◽  
Roshan Makkar
Keyword(s):  
Optical Coherence Tomography ◽  
Fourier Transform ◽  
Optical Coherence ◽  
Cross Sectional ◽  
Image Construction ◽  
Data Points ◽  
Maximum Penetration ◽  
The Fourier Transform ◽  
Processing Techniques

Optical coherence tomography (OCT) is an evolving medical imaging technology that offers in vivo cross-sectional, sub-surface images in real-time. OCT has become popular in the medical as well as non-medical fields. The technique extensively uses for food industry, dentistry, dermatology, and ophthalmology. The technique is non-invasive and works on the Michelson interferometry principle, i.e., dependent on back reflections of the signal and its interference. The objective is to develop an algorithm for signal processing to construct an OCT image and then to enhance the quality of the image using image processing techniques like filtering. The image construction was primarily based on the Fourier transform (FT) of the dataset obtained by data acquisition. This FT could be performed rapidly with the extensively used algorithm of fast Fourier transform (FFT). The depth-wise information could be extracted from each A-scan, i.e., axial scan and also the B-scan was obtained from the A-scan to see the structure of sample. The maximum penetration depth achieved with proposed system was 2.82mm for 1024 data points. First and second layer of leaf were getting at thickness of 1mm and 1.6mm, respectively. A-scans for Human fingertip gave its first, second and third layer was at a thickness of 0.75mm, 0.9mm and 1.6mm, respectively. A-scans for foam sheet gave its first, second and third layer was at a thickness of 0.6mm, 0.75mm, and 0.85mm, respectively.

scholarly journals Solar observation with the Fourier transform spectrometer I : Preliminary results of the visible and near-infrared solar spectrum

2021 ◽  
Vol 21 (10) ◽  
pp. 267
Author(s):  
Xian-Yong Bai ◽  
Zhi-Yong Zhang ◽  
Zhi-Wei Feng ◽  
Yuan-Yong Deng ◽  
Xing-Ming Bao ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Spectral Resolution ◽  
Near Infrared ◽  
Solar Spectrum ◽  
High Spectral Resolution ◽  
Inversion Method ◽  
Fourier Transform Spectrometer ◽  
Solar Observation ◽  
The Fourier Transform ◽  
Sun Light

Abstract The Fourier transform spectrometer (FTS) is a core instrument for solar observation with high spectral resolution, especially in the infrared. The Infrared System for the Accurate Measurement of Solar Magnetic Field (AIMS), working at 10–13 μm, will use an FTS to observe the solar spectrum. The Bruker IFS-125HR, which meets the spectral resolution requirement of AIMS but simply equips with a point source detector, is employed to carry out preliminary experiment for AIMS. A sun-light feeding experimental system is further developed. Several experiments are taken with them during 2018 and 2019 to observe the solar spectrum in the visible and near infrared wavelength, respectively. We also proposed an inversion method to retrieve the solar spectrum from the observed interferogram and compared it with the standard solar spectrum atlas. Although there is a wavelength limitation due to the present sun-light feeding system, the results in the wavelength band from 0.45–1.0 μm and 1.0–2.2 μm show a good consistency with the solar spectrum atlas, indicating the validity of our observing configuration, the data analysis method and the potential to work in longer wavelength. The work provided valuable experience for the AIMS not only for the operation of an FTS but also for the development of its scientific data processing software.

Comparison of Numerical and Experimental Strain Measurements of a Composite Panel Using Image Decomposition

2011 ◽  
Vol 70 ◽  
pp. 63-68 ◽  
Author(s):  
Christopher M Sebastian ◽  
Eann A Patterson ◽  
Donald Ostberg
Keyword(s):  
Fourier Transform ◽  
Image Decomposition ◽  
Image Correlation ◽  
Composite Panel ◽  
Data Sets ◽  
Full Field ◽  
Strain Measurements ◽  
Data Points ◽  
Optical Strain ◽  
The Fourier Transform

Image decomposition is used to address the problem of accurately and concisely describing the strain in an inhomogeneous composite panel that is bolted to a vehicle structure. In-service, the composite panel is subject to structural loads from the vehicle which can cause unintended damage to the panel. Finite element simulations have been performed with the plan to establish their fidelity using full-field optical strain measurements obtained using digital image correlation. A methodology is presented based on using orthogonal shape descriptors to decompose the data-rich maps of strain into information-preserved data sets of reduced dimensionality that facilitate a quantitative comparison of the computational and experimental results. The decomposition is achieved employing the Fourier transform followed by fitting Tchebichef moments to the maps of the magnitude of the Fourier transform. The results show that this approach is fast and reliably describes the strain fields using less than fifty moments as compared to the thousands of data points in each strain map.

2-D Hartley transforms

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 262-267 ◽  
Author(s):  
N. Sundararajan
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Digital Signal Processing ◽  
Digital Signal ◽  
Physical Significance ◽  
The Real ◽  
Hartley Transform ◽  
Real Domain ◽  
Hartley Transforms ◽  
The Fourier Transform

Two different versions of kernels associated with the 2-D Hartley transforms are investigated in relation to their Fourier counterparts. This newly emerging tool for digital signal processing is an alternate means of analyzing a given function in terms of sinusoids and is an offshoot of Fourier transform. Being a real‐valued function and fully equivalent to the Fourier transform, the Hartley transform is more efficient and economical than its progenitor. Hartley and Fourier pairs of complete orthogonal transforms comprise mathematical twins having definite physical significance. The direct and inverse Hartley transforms possess the same kernel, unlike the Fourier transform, and hence have the dual distinction of being both self reciprocal and having the convenient property of occupying the real domain. Some of the properties of the Hartley transform differ marginally from those of the Fourier transform.

scholarly journals One Technique to Enhance the Resolution of Discrete Fourier Transform

Electronics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 330 ◽  
Author(s):  
Ivan Kanatov ◽  
Dmitry Kaplun ◽  
Denis Butusov ◽  
Viacheslav Gulvanskii ◽  
Aleksander Sinitca
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Digital Signal ◽  
Analysis Tool ◽  
Signal Phase ◽  
Practical Applications ◽  
Advantages And Disadvantages ◽  
Engine Vibration ◽  
Window Functions ◽  
The Fourier Transform

Discrete Fourier transform (DFT) is a common analysis tool in digital signal processing. This transform is well studied and its shortcomings are known as well. Various window functions (e.g., Hanning, Blackman, Kaiser) are often used to reduce sidelobes and to spread the spectrum. In this paper, we introduce a transformation that allows removing the sidelobes of the Fourier transform and increasing the resolution of the DFT without changing the time sample. The proposed method is based on signal phase analysis. We give the comparison of the proposed approach with known methods based on window functions. The advantages and disadvantages of the proposed technique are explicitly shown. We also give a set of examples illustrating the application of our technique in some practical applications, including engine vibration analysis and a short-range radar system.

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